Dynamical Systems - 8/4/10

Today, we were treated to a quick introduction to dynamical systems, an area of math which focuses on repeated iteration of maps to study long-term or asymptotic behavior associated with this iteration.

We started by taking a simple example. Consider a real valued function f from R to R. Then consider the sequence of points (x, f(x), f(f(x)) ....) We call this sequence the orbit of the point x.

Let us take f(x) = kx(1-x), which is the equation for a quadratic function. Also consider the identity map, g(x) = x.

For now, we take k >0.

Depending on our specific value of k, we have a parabola which intersects the g(x) either once or twice.

We are interested in the fixed points of f. As the name implies, these are points where the map f leaves a point in its domain fixed, or in symbols: f(x1) = x1.

We wish to study asymptotic behavior of functions. To do this, we pick a point x on the real line and first apply the function. This gives us a new point f(x). Then we find f(f(x)). Since we have the identity map also drawn in (in this case, g(x)), this process is applying f, moving horizontally to g, moving vertically to f, and so on.

We see that repeating this process, we notice two general patterns: In some situations, we may continue onward towards positive or negative infinity in an unbounded fashion. However, for other sections of the real line, we see that there is a convergence towards certain fixed points.

We call a fixed point x attracting if the magnitude of its derivative at x is less than 1 and we say a fixed point is repelling if the magnitude of its derivative at x is greater than 1.

We see then that the derivative of the function at these fixed points greatly affects this convergence behavior.

Afterwards, we moved on to dynamics of complex functions which map the complex plane to itself.

We first took f: C -> C (complex plane to itself) given by f(z) = z^2 + c for some complex number c.

If we let c = 0, then points inside the unit disk spiral towards the origin. Points outside the unit disk spiral to infinity. We see that points on the unit circle itself get mapped to other points on the unit circle (their length is preserved, but they rotate.) Thus the mapping leaves the unit circle invariant. Furthermore, we see that iterates of the map can map a small section of the unit disk onto the whole unit disk.

We now consider the effect of varying our complex number c. For c very small, we obtain a similar picture: the interior of the curve spirals inward to the center, the curve itself is invariant, and the outer regions escape to infinity.

For very large x, most points escape outward to infinity.

We then construct the Mandelbrot set as the set of points c such that the n-th iterate of the map f does not go to infinity when evaluated at the point c itself for technical reasons.

We take all parameters c and iterate at the point c itself.

Similar to our notion of fixed points, we say a point z in the complex plane is periodic with a period n if the n-th iterate of f applied at z = z.

We concluded by seeing various pictures of the Mandelbrot set and fractals constructed through similar dynamical systems in the complex plane.